Does this mean that we use peux-je when talking solely about ability eg "Peux-je le faire ? Oui, je peux.", but we use puis-je when asking for permission? As a child I remember being regularly reproached with "'May I?', not 'Can I?'!" Is the same true with French? Is puis-je "sloppy" French?

Many thanks for the clarification. Puis-je is included as an example in the Tips & Notes on Questions, but there is no explanation on its usage. Thanks for filling the hole. If I ever get to grips with this language, it'll be as much through your efforts as mine...

There's no subjunctive to be found here. The subjunctive virtually always uses "que" before it. Additionally, "may I" would be conditional, not subjunctive. "Pourrais-je" would be more along the lines of what you would expect.

In Australia we have different cards which we can use to pay for things:

Credit card - a bank account, which uses the bank's money to pay for the goods.

Debit card - a card which acts like a credit card, but uses your money from your bank account (it works like a VISA card, and you can use Paywave)

Savings (EFTPOS) card - a card which uses your money, but does not act like a credit card (you can't use it like a VISA card)

Rewards card - you accumulate "points" on your card from purchases, and when you have enough points, you can redeem them at the card vendor's nominated outlets. Often associated with airlines, large department stores, etc.

Gift card - various supermarkets / shopping centres have these - you buy a card with a nominated amount (eg. $20, $50 etc.) or nominate the value you want to put on the card, and these can be used at nominated locations.

If you waved any of these at the person you're trying to pay, they'd immediately recognise it and tell you whether you could use it or not (so you don't have to say "credit card", "gift card" etc.)

If you didn't flash the card at them, you'd have to specify "Can I use my credit card?" / "Can I use my gift card?" etc.

My question is whether or not there is another French verb that differentiates "May I" from "Can I" ?
My mother would have corrected me in my (American) English if I had asked her "Can I ride my bike?" rather than "May I ride my bike?"
Her point was that in English "can" implies the ability, whereas "may" was the polite way to ask her permission.
I translated the French sentence with "May I pay with my card?" Duo gave me the green check. I preferred "May I?" here because I imagined I was speaking to a shopkeeper in France. "Can I?" would then be determined by the issuer of the card with either an "accepted" or "declined".
Well, in spite of myself, I have descended into creating "clutter". I did read the other posts and don't think my question was addressed previously by others.
Latin was my favorite language in high school. It really loved it.

I think I understand. Pouvoir, though it is more often translated to English with the "can", has nothing to do with ability. It is for either asking permission, or questioning someone's availability to do whatever in a polite enough way. Am I on the right track?
Thanks for the options above regarding capability.
BTW - - questioning my own abilities happens quite often in my head.

My French professor told me "puis-je" and "pourrais-je" are considered archaic and aren't used anymore, and that she got made fun of for using them in France her first time visiting. Can you confirm this?

Nothing really unless one is going to go strictly pedantic. The expression would certainly be understood as "credit card". In context, only a stringent, literal approach would not understand that "carte" refers to carte de crédit.